An Introduction to Functional Analysis in Computational Mathematics: An Introduction

Presents the basics of functional analysis, as well as elements of variational equations (on the basis of bi-linear forms), including the Vishik-Lax-Milgram theorem and of generalized solutions of eliptic problems. In terms of functional analysis, such problems of computational mathematics are considered as extremal of problems of approximation theory of various types - the theory of numerical integration, variational methods of minimization of quadratic functional, and Galerkin and Ritz methods of finding solutions to operator equations. Also covered are: the general theory of iteration methods, Chebyshen iteration methods, composition method, along with some elements of nonlinear analysis. Sobolev spaces and embedding theorems are also introduced.

Table of Contents:
1. Functional Spaces and Problems in the Theory of Approximation.- 1. Metric Spaces.- 2. Compact Sets in Metric Spaces.- 3. Statement of the Main Extremal Problems in the Theory of Approximation. Main Characteristics of the Best Approximations.- 4. The Contraction Mapping Principle.- 5. Linear Spaces.- 6. Normed and Banach Spaces.- 7. Spaces with an Inner Product. Hilbert Spaces.- 8. Problems on the Best Approximation. Orthogonal Expansions and Fourier Series in a Hilbert Space.- 9. Some Extremal Problems in Normed and Hilbert Spaces.- 10. Polynomials the Least Deviating from Zero. Chebyshev Polynomials and Their Properties.- 11. Some Extremal Polynomials.- 2. Linear Operators and Functionals.- 1. Linear Operators in Banach Spaces.- 2. Spaces of Linear Operators.- 3. Inverse Operators. Linear Operator Equations. Condition Measure of Operator.- 4. Spectrum and Spectral Radius of Operator. Convergence Conditions for the Neumann Series. Perturbations Theorem.- 5. Uniform Boundedness Principle.- 6. Linear Functionals and Adjoint Space.- 7. The Riesz Theorem. The Hahn-Banach Theorem. Optimization Problem for Quadrature Formulas. The Duality Principle.- 8. Adjoint, Selfadjoint, Symmetric Operators.- 9. Eigenvalues and Eigenelements of Selfadjoint and Symmetric Operators.- 10. Quadrature Functionals with Positive Definite Symmetric or Symmetrizable Operator and Generalized Solutions of Operator Equations.- 11. Variational Methods for the Minimization of Quadrature Functionals.- 12. Variational Equations. The Vishik-Lax-Milgram Theorem.- 13. Compact (Completely Continuous) Operators in Hilbert Space.- 14. The Sobolev Spaces. Embedding Theorems.- 15. Generalized Solution of the Dirichlet Problem for Elliptic Equations of the Second Order.- 3. Iteration Methods for the Solution of Operator Equations.- 1. General Theory of Iteration Methods.- 2. On the Existence of Convergent Iteration Methods and Their Optimization.- 3. The Chebyshev One-Step (Binomial) Iteration Methods.- 4. The Chebyshev Two-Step (Trinomial) Iteration Method.- 5. The Chebyshev Iteration Methods for Equations with Symmetrized Operators.- 6. Block Chebyshev Method.- 7. The Descent Methods.- 8. Differentiation and Integration of Nonlinear Operators. The Newton Method.- 9. Partial Eigenvalue Problem.- 10. Successive Approximation Method for Inverse Operator.- 11. Stability and Optimization of Explicit Difference Schemes for Stiff Differential Equations.